Menu location: Analysis_Exact_Odds Ratio CI.
Odds = probability / (1 - probability) therefore odds can take on any value between 0 and infinity whereas probability may vary only between 0 and 1. Odds and log odds are therefore better suited than probability to some types of calculation.
Odds ratio (OR) is related to risk ratio (RR, relative risk):
RR = (a / (a+c)) / (b / (b+d))
When a is small in comparison to c and b is small in comparison to d (i.e. relatively small numbers of outcome positive observations or low prevalence) then c can be substituted for a+c and d can be substituted for d+b in the above. With a little rearrangement this gives the odds ratio (cross ratio, approximate relative risk):
OR = (a*d)/(b*c).
OR can therefore be related to RR by:
RR = 1/(BR+(1-BR)/OR)
..where BR is the baseline (control) response rate; BR can be estimated by b/(b+d) if not known from larger studies.
This function uses an exact method to construct confidence limits for the odds ratio of a fourfold table (Martin and Austin, 1991). The Fisher limits complement Fisher's exact test of independence in a fourfold table, for which one and two sided probabilities are provided here. Mid-P values are also given.
Please note that this method will take a long time with large numbers.
Observed frequencies should be entered as a standard fourfold table:
|feature present||feature absent|
sample estimate of the odds ratio = (a*d)/(b*c)
From Thomas (1971).
The following data look at the criminal convictions of twins in an attempt to investigate some of the hereditability of criminality.
To analyse these data in StatsDirect select Odds Ratio Confidence Interval from the Exact Tests section of the analysis menu. Choose the default 95% two sided confidence interval.
For this example:
Confidence limits with 2.5% lower tail area and 2.5% upper tail area two sided:
Observed odds ratio = 25
Conditional maximum likelihood estimate of odds ratio = 21.305318
Exact Fisher 95% confidence interval = 2.753383 to 301.462338
Exact Fisher one sided P = 0.0005, two sided P = 0.0005
Exact mid-P 95% confidence interval = 3.379906 to 207.270568
Exact mid-P one sided P = 0.0002, two sided P = 0.0005
Here we can say with 95% confidence that one of a pair of identical twins who has a criminal conviction is between 2.75 and 301.5 times more likely than non-identical twins to have a convicted twin.